Quantum and Classical Algorithms for Bounded Distance Decoding

TL;DR

This paper compares quantum and classical algorithms for bounded distance decoding in lattices, showing that classical algorithms can match quantum performance in certain cases, clarifying the computational landscape.

Contribution

It provides a detailed proof that classical algorithms can solve certain BDD problems as efficiently as quantum algorithms, resolving a recent debate.

Findings

Classical algorithms can solve $ ext{BDD}$ problems as efficiently as quantum algorithms.

The paper offers detailed proofs of classical algorithms matching quantum results.

Clarifies the computational boundaries between quantum and classical approaches.

Abstract

In this paper, we provide a comprehensive overview of a recent debate over the quantum versus classical solvability of bounded distance decoding (BDD). Specifically, we review the work of Eldar and Hallgren [EH22], [Hal21] demonstrating a quantum algorithm solving $\lambda_1 2^{-\Omega(\sqrt{k \log q})}$-BDD in polynomial time for lattices of periodicity $q$, finite group rank $k$, and shortest lattice vector length $\lambda_1$. Subsequently, we prove the results of [DvW21a], [DvW21b] with far greater detail and elaboration than in the original work. Namely, we show that there exists a deterministic, classical algorithm achieving the same result.